The Blues dropped points at home eight times in 19 Premier League games last season.
Morata told Chelsea TV: "Other teams need to fight a lot to get something here.
"This is the message we need to send to other teams. Here is our home and it's difficult to get something here.
"We are very happy because we deserved the victory. We need to improve little things, but we played good with the ball. A very important three points for us."
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Morata scored just his second goal in 16 league appearances since Boxing Day, after Pedro had given Chelsea an early lead.
But Arsenal fought back through goals from Henrikh Mkhitaryan and Alex Iwobi only for Eden Hazard to provide the winner for Marcos Alonso.
Morata added: "I only want to win. Now we can go home happy because we have the three points, another victory.
"The supporters are always happy if we win. For sure we don't want a lot of goals in our goal, but if we win all the team and all the supporters are happy."
Arsenal boss Unai Emery insisted his decision to substitute Granit Xhaka at half-time was tactical, partly due to the Swiss midfielder's first-half booking.
Unai Emery is still waiting for his first win (Arsenal FC via Getty)
He also downplayed the replacement of Mesut Ozil.
Emery said: "I am happy with Mesut. He worked. I decided to change because we needed more in the middle. We were defending very deep."
On Xhaka, Emery said: "We don't take risks with this yellow card."